#8994713

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#53994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,8)(3,9,10,4)(5,7,13,14)(11,15,16,12)(17,18,22,27)(19,23,30,32)(20,25,29,24)(21,26,28,31);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,30)(18,23,27,32)(20,28,29,21)(24,26,25,31)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(3,10)(4,9)(5,13)(7,14)(17,20,22,29)(18,24,27,25)(19,21,30,28)(23,31,32,26)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,3,6,10)(2,4,8,9)(5,11,13,16)(7,12,14,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,27)(26,32)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,22,28)(18,26,27,31)(19,29,30,20)(23,24,32,25)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5,6,13)(2,7,8,14)(3,11,10,16)(4,12,9,15)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901909);